On Analytic Formulas of Feynman Propagators in Position Space *

CPC(HEP & NP), 2010, 34(10): 1576–1582 Chinese Physics C Vol. 34, No. 10, Oct., 2010

On analytic formulas of Feynman propagators in position space *

ZHANG Hong-Hao(Ü÷Ó)1;1) FENG Kai-Xi(¾mU)1 QIU Si-Wei(£d)1 ZHAO An(ëS)1 LI Xue-Song(oÈt)2

1 School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China 2 Science College, Hunan Agricultural University, Changsha 410128, China

Abstract We correct an inaccurate result of previous work on the Feynman propagator in position space of a free Dirac field in (3 + 1)-dimensional spacetime; we derive the generalized analytic formulas of both the scalar Feynman propagator and the spinor Feynman propagator in position space in arbitrary (D + 1)- dimensional spacetime; and we further find a recurrence relation among the spinor Feynman propagator in (D+1)-dimensional spacetime and the scalar Feynman propagators in (D+1)-, (D−1)- and (D+3)-dimensional spacetimes. Key words Feynman propagator, Klein-Gordon field, Dirac field, arbitrary dimensional spacetime PACS 11.10.Kk, 03.70.+k, 02.30.Gp

1 Introduction language of quantum field theory, the Feynman propagator in position space can be physically understood Although one cannot adopt the extreme view that as the energy due to the presence of the two external the set of all Feynman rules represents the full the- sources located at two different points in space and ory of quantized fields, the approach of the Feynman acting on each other (See Ref. [1] Chapt.I.4). graphs and rules plays an important role in pertur- It has been mentioned in Refs. [1–8] that in (3+1)- bative quantum field theories. For a generic quan- dimensional spacetime, after integrating over momentum theory involving interacting fields, the set of its tum, the Feynman propagator of a free Klein-Gordon Feynman rules includes some vertices and propaga- field can be expressed in terms of Bessel or modified tors. While for a free field theory there is only one Bessel functions, which depends on whether the sep- graph, that is, the Feynman propagator, in the set aration of two spacetime points is timelike or space- of its Feynman rules. Thus the Feynman propagator like. By changing variables to hyperbolic functions describes most, if not all, of the physical contents of and using the integral representation of the Hankel the free field theory. It is true that the real world is function of the second kind, the authors of Ref. [2] not governed by any free field theory. However, this derived the full analytic formulas of the Feynman kind of theory is the basis of a perturbatively inter- propagors of free Klein-Gordon and Dirac fields in acting field theory, and the issues of a free theory are (3+1)-dimensional spacetime. The expressions of the usually in the simplest situation and thus their study Feynman propagators of a free Klein-Gordon field in will be attempted at the first step of investigations. (1+1)- and that in (2+1)-dimensional spacetime can In statistical field theory, the Feynman propagator is be found in Refs. [9] and [10], respectively. How- usually called the correlation function. We need to ever, in Ref. [2] the expression for the Feynman prop- know its formula in position space to figure out the agator of a Dirac spinor field is inaccurate, since critical exponent η and the correlation length ξ, which there is at least a redundant term in their results. is related to another exponent ν. In the path integral In this paper we will show that the term actually

Received 20 November 2009, Revised 21 January 2010 * Supported by Specialized Research Fund for Doctoral Program of Higher Education (SRFDP) (200805581030) and Sun Yet- Sen University Science Foundation and Fundamental Research Funds for the Central Universities 1) E-mail: [email protected] ©2010 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd No. 10 ZHANG Hong-Hao et al: On analytic formulas of Feynman propagators in position space 1577 vanishes and we will give the correct expression. Fur- Thus, we can obtain the analytic formula of the Feyn- thermore, we will generalize the results of previous man propagator in position space by integrating over work and derive the full analytic formulas of the Feyn- the D-dimensional momentum in the expression of man propagators in position space of, respectively, D( t ,~x). This integral in D-dimensional Euclidean | | the Klein-Gordon scalar and the Dirac spinor in arbi- space can be evaluated by changing the variables from trary (D + 1)-dimensional spacetime. Eventually we Cartesian coordinates to spherical coordinates. Since will find an interesting recurrence relation between the angular integral parts look a little different in the spinor Feynman propagator in (D+1)-dimensional (1+1)-, (2+1)- and general (D +1)-dimensional (for spacetime and the scalar Feynman propagators in D > 3) spacetimes, in order to be more careful in our (D + 1)- and alternate-successive dimensional space- derivation, let us consider these situations case by time. case. Eventually we will show that the general result This paper is organized as follows. In Section 2, of (D+1)-dimensional spacetime holds for D = 1,2 as we will briefly review the derivation of the analytic well. formulas of the Feynman propagators of a free Klein- 2.1 (1+1)-dimensional spacetime Gordon field in (1+1)- and (2+1)-dimensional spacetime, and we will compute, once and for all, the When the spatial dimension D = 1, Eq. (3) be- scalar Feynman propagator in (D + 1)-dimensional comes +∞ spacetime. After completion of this work, we became dp 1 −i(E|t|−pr) DF(t,r) = π e , (4) aware of the analytic formula of the scalar Feynman Z−∞ 2 2E propagator in position space in arbitrary dimensional with E = √p2 +m2. Using the substitution E = m spacetime which is also given in Ref. [11]. We will cosh η, p = m sinh η (with < η < ) in the above demonstrate that our result exactly agrees with that −∞ ∞ integral, we have of Ref. [11] in this section. In Section 3 we will make 1 +∞ use of the obtained formula of the scalar Feynman D (t,r) = dηe−im(|t| cosh η−r sinhη). (5) F π Z propagator to compute the expression of the spinor 4 −∞ Feynman propagator in (D+1)-dimensional spacetime Due to the Lorentz invariance, the Feynman propa- and obtain a recurrence relation. We will also com- gator can depend only on the interval x2 t2 r2. ≡ − pare our result for D = 3 with that of Ref. [2], and we If the interval is timelike, x2 > 0, we can make a will show that one additional term in Ref. [2] actually Lorentz transformation such that x is purely in the has no contribution and that the method they used time-direction, x0 = θ(t)√t2 r2 i, r = 0. Note − − to prove the term nonzero was inappropriate. The that √s is not a single-valued-function of s and here last section is devoted to conclusions. and henceforth the cut line in the complex plane of s is chosen to be the negative real axis. The negative infinitesimal imaginary part, i, is because of 2 Feynman propagator of Klein- − the Feynman description of the Wick rotation, i.e., Gordon theory x2 x2 i in position space and correspondingly → − k2 k2 +i in momentum space. Thus, → Following the notation of Refs. [1, 4], we write +∞ 1 −im√t2−r2−i cosh η the Feynman propagator of a free Klein-Gordon field DF(t,r) = π dηe 4 Z−∞ φ(x) φ(t,~x) in (D+1)-dimensional spacetime as the i ≡ = H (2)(m√t2 r2 i), (6) time-ordered two-point correlation function, −4 0 − − (2) DF(x) 0 T φ(x)φ(0) 0 where H0 (x) is the Hankel function of the second ≡ h | | i kind, and where we have used the identity in # 3.337 = θ(x0)D(x)+θ( x0 )D( x) , (1) − − of Ref. [12], +∞ with the unordered two-point correlation function, −iβ cosh η π (2) π dηe = i H0 (β), ( < argβ < 0). (7) Z−∞ − − dDp~ 1 −i(Ep~t−p~·~x) 2 D(x) 0 φ(x)φ(0) 0 = D e . (2) Likewise, if the interval is spacelike, x < 0, we have ≡ h | | i Z (2π) 2Ep~ +∞ 1 im√r2−t2+i sinhη Combining the above two equations gives DF(t,r) = π dηe 4 Z−∞ dDp~ 1 −i(Ep~|t|−p~·~x) 1 D (t,~x) = D( t ,~x) = e . (3) 2 2 F π D = K0(m√r t +i), (8) | | Z (2 ) 2Ep~ 2π − 1578 Chinese Physics C (HEP & NP) Vol. 34

1 2 where K0(x) is the modified Bessel function, and + θ( x2) e−m√−x +i. (16) − 4π√ x2 +i where we have used the identity in # 3.714 of − Ref. [12], Eqs. (13) and (16) agree well with the results of pre- ∞ π π vious work [9, 10]. dη cos(β sinhη) = K0(β), < argβ < . (9) Z0 − 2 2 2.3 (D+1)-dimensional spacetime (for D > 3) It is worthwhile noting that the i description as- − sures the proper phase angles of √x2 i in Eq. (6) Now, let us proceed to compute the scalar Feyn- − and √ x2 +i in Eq. (8), respectively, so that the man propagator in (D+1)-dimensional spacetime (for − mathematical identities (7) and (9) can be applied D > 3). Since the method we use in the following to figure out these two expressions. In the limit of differs from that used in Ref. [2], let us wait to see x2 0, both Eqs. (6) and (8) are divergent and are whether the results from these two approaches are → approaching consistent or not. Changing the variables from Carte- sian coordinates to spherical coordinates, Eq. (3) be- i (2) 2 1 1 lim H0 (m√x i) ln , (10) x2→0 −4 − ∼ 4π x2 i comes − D−1 π π 2 1 2 D−2 1 1 1 DF(t,r) = sin θdθ 2 π D lim K0(m√ x +i) ln , (11) (2 ) D 1 Z0 x2→0 2π − ∼ 4π x2 i Γ − − 2 which are of the same form and do not depend on the mass m, and which can be recognized as the scalar ∞ pD−1 dp e−i(E|t|−pr cos θ), (17) Feynman propagator on the lightcone. The above ex- ×Z0 2E pression is indeed the Feynman propagator of a mass- with E = √p2 +m2. To evaluate the above inte- less scalar field, gral, we need to figure out the angular integral π 1 1 D−2 ipr cos θ DF(x) = ln , for m = 0. (12) dθ sin θe . From the formula # 3.387 of 4π x2 i Z − 0 In summary, the scalar Feynman propagator in posi- Ref. [12], tion space may be written in a compact way as 1 dx(1 x2)ν−1eiµx 2 i (2) 2 Z−1 − DF(x) = θ(x ) H0 (m√x i) − 4 − 1 ν− 2 π 2 1 = √ Γ (ν)Jν− 1 (µ), (Re ν > 0), (18) + θ( x2) K (m√ x2 +i), (13) µ 2 − 2π 0 − where the theta function θ(x) is defined as we can easily find that π k x 2 1 , (x > 0) k ipr cos θ π 2 k +1 θ(x) = δ(y)dy = (14) dθ sin θe = √ Γ J k (pr). Z pr 2 2 Z−∞ ( 0 , (x < 0) 0 (19) and the value of θ(x = 0) depends on whether the ar- Then, substituting Eq. (19) with k = D 2 into − gument x is approaching 0 from the positive or neg- Eq. (17), we obtain ative real axis, that is, θ(0+) = 1 and θ(0−) = 0. ∞ D 2 1 p −iE|t| DF(t,r) = dp J D (pr)e , 2.2 (2+1)-dimensional spacetime D D −1 2 −1 2(2π) 2 r 2 Z0 E In (2 + 1)-dimensional spacetime, the Feynman (20) propagator of a free scalar field is which, by changing the variable of integration to 2π ∞ x = E/m, leads to 1 p −i(E|t|−pr cos θ), DF(t,r) = dθ dp e (15) D ∞ (2π)2 Z Z 2E m 2 1 D 0 0 2 2 ( 2 −1) DF(t,r) = D D dx(x 1) π 2 2 −1 Z with E = √p2 +m2. Using a similar calculation proce- 2(2 ) r 1 −

dure, we can obtain the analytic formula of the scalar 2 −im|t|x J D (mr√x 1)e . (21) Feynman propagator in (2+1)-dimensional spacetime × 2 −1 − as follows, To compute the above integral, we can make the analytical continuation of the 2nd formula of # 6.645 of 2 i −im√x2−i DF(x) = θ(x ) − e 4π√x2 i Ref. [12], − No. 10 ZHANG Hong-Hao et al: On analytic formulas of Feynman propagators in position space 1579

∞ 1 2 1 1 2 2 ν −αx 2 ν 2 2 − 2 ν− 4 2 2 dx(x 1) e Jν (β√x 1) = β (α +β +i) Kν 1 α +β +i (22) π + 2 Z1 − − r and obtain the following identity: p

2 1 1 ν 2 2 − 2 ν− 4 2 2 b (b a +i) Kν 1 (√b a +i), (b > a > 0) ∞ π + 2 1 − − 2 ν −iax 2 r dx(x 1) 2 e Jν (b√x 1) =  . (23) 2(ν+1) Z1 − −  π ( i)  ν (2) 2 2  b H 1 √a b i , (a > b > 0) − ν 1 ν+ 2 √ 2 2 + 2 2 − − r ( a b i)  − − D  Substituting Eq. (23) (with ν = 1, a =mt, b = mr) into Eq. (21), we obtain 2 − D−1 D 2 ( i) m (2) 2 2 2 2 DF(t,r) = D+3 D−1 D−1 H D−1 (m√t r i), (for t r > 0), (24) 2 2 π 2 (t2 r2 i) 4 2 − − − − − D−1 2 m 2 2 − 2 2 DF(t,r) = D+1 D−1 K D 1 (m√r t +i), (for r t > 0). (25) (2π) 2 (r2 t2 +i) 4 2 − − − In the limit of x2 0, the above two formulas are to Eqs. (13),(16), respectively. Therefore, in the fol- → approaching a common asymptotic expression, that lowing we will use Eq. (27) to describe the scalar is, the scalar Feynman propagator on the lightcone, Feynman propagator in (D + 1)-dimensional space- > D 1 time for D 1. The shapes of the scalar Feynman Γ − D−1 propagators with spacelike or timelike separations in 2 1 2 2 DF(x) D+1 2 , (for x 0), different dimensional spacetime are shown in Figs. 1 ∼ π 2 − x i → 4 − (26) and 2, respectively. The figures show that in any di- which is indeed the exact formula of the Feynman mensional spacetime, the spacelike propagation am- propagator of a massless scalar field. In summary, the plitude is dominated by the exponential decay, while full analytic expression of the scalar Feynman propa- the timelike propagation amplitude behaves as the gator in (D +1)-dimensional spacetime is given by damped oscillation; and in both cases the more the dimension of spacetime, the more rapidly the propa- DF(x) gation amplitude decreases. D−1 D 2 2 ( i) m (2) 2 = θ(x ) D+3 −D−1 D−1 H D−1 (m√x i) 2 2 π 2 (x2 i) 4 2 − − D−1 2 2 m +θ( x ) D+1 D−1 − (2π) 2 ( x2 +i) 4 − 2 K D−1 (m√ x +i). (27) × 2 − In particular, taking D = 3, it follows from the above equation that

2 im (2) 2 DF(x) = θ(x ) H (m√x i) 8π√x2 i 1 − − 2 m 2 +θ( x ) K1(m√ x +i),(28) − 4π2√ x2 +i − − which is consistent with the results in (3 + 1)- dimensional spacetime of Ref. [2]. Moreover, Eq. (27) > holds not only for D 3 but also for D = 1,2. Noting Fig. 1. The scalar Feynman propagator DF(0,r) the facts that with spacelike separation r in (D + 1)-dim- π ensional spacetime, where we have set the (2) 2 −ix −x H 1 (x) = i e , K 1 (x) = e , (29) mass parameter m = 1, and the solid-line cor- 2 πx 2 2x r r responds to D = 1, while the dashed-lines, it can easily be veriﬁed that if the spatial dimen- from long to short, correspond to D = 2,3,4,5, sion is taken to be D = 1,2, Eq. (27) will reduce respectively. 1580 Chinese Physics C (HEP & NP) Vol. 34

Proof: Let the cut line in the complex plane be the negative real axis, so that the argument range of the variable x of the complex function f(x) = √x must be ( π/2, π/2). First of all, let us show that − √ x2 +i = i√x2 i (32) − − holds in the spirit of analytical continuation:

1 1) If x2 > 0, we have √x2 i = x2 2 . On the other − π | | 1 π 1 hand, √ x2 +i = x2 ei = x2 2 ei 2 = i x2 2 . − | | | | | | Thus Eq.(32) holds. p

1 2) If x2 < 0, we have √ x2 +i = x2 2 . On the − π| | 1 π other hand, √x2 i = x2 e−i = x2 2 e−i 2 = 1 − | | | | i x2 2 . Thus Eq.(32) holds as well. − | | p Now, to prove Eq. (31), it is suﬃcient to prove im m H (2)(mz) = K (imz), (33) 8πz 1 4π2iz 1 where the argument of z is Argz ( π/2, 0), and ∈ − which is equivalent to π K (iz) = H (2)(z), (34) 1 − 2 1 which can be checked from the deﬁnition of the

function Kν (z). Therefore, we complete our proof of Eq. (31). Since any function f(x) can be written as f(x) = θ(x)f(x)+θ( x)f(x), we ﬁnally obtain −

2 im (2) 2 DF(x) = θ(x ) H1 (m√x i) Fig. 2. The real and imaginary parts of the 8π√x2 i − − scalar Feynman propagator DF(t,0) with 2 m 2 timelike separation t in (D + 1)-dimensional +θ( x ) K1(m√ x +i) − 4π2√ x2 +i − spacetime, where we have set the mass param- − eter m = 1, and the solid-line corresponds to im = H (2)(m√x2 i) D = 1, the long dashed-line D = 2 and the 8π√x2 i 1 − short dashed-line D = 3. − m 2 = K1(m√ x +i), (35) After completion of this work, we became aware of 4π2√ x2 +i − − the analytic formula of the scalar Feynman propaga- which demonstrates the equivalence of our result and tor in position space in arbitrary dimensional space- that of Ref. [11]. time which is also presented in Ref. [11],

D−1 D 2 ( i) m (2) 2 3 Feynman propagator of Dirac DF(x) = D+3 −D−1 D−1 H D−1 (m√x i). 2 2 π 2 (x2 i) 4 2 − theory − (30) In the following, let us demonstrate that Eq. (27) is In this section, let us calculate the analytic for- equivalent to Eq. (30). That is, we need to prove the mula of the Feynman propagator in position space two parts in Eq. (27) are actually equal in the spirit of a free Dirac spinor ﬁeld in (D + 1)-dimensional of analytical continuation. For simplicity of the no- spacetime. Since we have obtained the exact expres- tation, let us take D = 3 as an example and show sion of the scalar Feynman propagator in any dimen- that sional spacetime, Eq. (27), it is straightforward to get im H (2)(m√x2 i) the expression of the spinor Feynman propagator by 8π√x2 i 1 − − means of the relation SF(x) = (i∂/ + m)DF(x). The m 2 = K1(m√ x +i). (31) result we obtain is 4π2√ x2 +i − − No. 10 ZHANG Hong-Hao et al: On analytic formulas of Feynman propagators in position space 1581

D+1 D−1 2 2 ( i) m x/ (2) 2 (2) 2 SF(x) = θ(x ) D+5− D−1 D+1 H D−3 (m√x i) H D+1 (m√x i) 2 2 π 2 (x2 i) 4 2 − − 2 − − D+1 2 2 im x/ − 2 2 +θ( x ) D+3 D+1 D+1 K D 3 (m√ x +i)+K D+1 (m√ x +i) − 2 2 π 2 ( x2 +i) 4 2 − 2 − − (D 1) ix/ − D (x)+mD (x) − 2 x2 i F F − D+1 D−1 2 ( i) m x/ (2) 2 (2) 2 = D+5− D−1 D+1 H D−3 (m√x i) H D+1 (m√x i) 2 2 π 2 (x2 i) 4 2 − − 2 − − (D 1) ix/ − D (x)+mD (x), (36) − 2 x2 i F F − µ where x/ γ xµ, and where we have used the following The reason the authors of Ref. [2] regarded the above ≡ (2) recurrence relations of the Hankel function Hν (x) term to be nonzero may come from the fact that they and the modified Bessel function K (x), had taken both x2 and x2 to be the absolute value ν − x2 simultaneously in their calculation. However, it is d 1 (2) (2) (2) | | 2 2 Hν (x) = Hν−1(x) Hν+1(x) , (37) obviously impossible that both x and x are equal dx 2 − − to x2 , even if x2 0, because x2 can only approach d 1 | | → Kν (x) = Kν (x)+Kν (x) . (38) zero from either the positive or the negative axis di- dx −2 −1 +1 rection, that is, in any case x2 and x2 always have − In particular, when the spatial dimension D = 3, opposite signs even if they are infinitesimal. Eq. (36) becomes Moreover, from Eq. (36) together with Eq. (27), m2x/ we obtain an interesting recurrence relation for the S (x) = θ(x2) H (2)(m√x2 i) F − 16π(x2 i) 0 − spinor Feynman propagator in (D + 1)-dimensional − h spacetime and the scalar Feynman propagators in H (2)(m√x2 i) − 2 − (D 1)-, (D+1)- and (D+3)-dimensional spacetime, im2x/i − +θ( x2) K (m√ x2 +i) as follows, π2 2 0 − 8 ( x +i) − ix/ − h S(D+1)(x) = +m D(D+1)(x) ix/ F 2 F +K (m√ x2 +i) D (x) − x i 2 − − x2 i F − i − im2x/ +mDF(x), (39) D(D−1)(x) −4π(x2 i) F which is a little different from the results of Ref. [2], − π (D+3) besides the less important factor of i owing to the +i x/DF (x), (41) convention that DF(x) here equals i∆F(x) there. The where the superscripts denote the spacetime dimen- essential difference between our results and those of sions of the respective physical quantities. The above Ref. [2] lies in the fact that there is an additional relation essentially stems from the fact that the Feyn- 2 term multiplied by δ(x ) in that book, which is pro- man propagator in any dimensional spacetime can be portional to Eq. (34) on page 80 of Ref. [2]. However, expressed in terms of Bessel and modified Bessel func- we find that this term is redundant, since its propor- tions, which has been proved in this paper. And it tional factor can be shown to vanish as follows, shows that the free Dirac theory and the free Klein- Gordon theories in alternate-successive dimensional 1 (2) 2 lim H1 (m√x i) x2→0 √x2 i − spacetime might be related to each other. − i H (2)( im√ x2 +i) −√ x2 +i 1 − − 4 Conclusions − 1 i 2 ∼ √x2 i π m√x2 i In this paper, we have pointed out and corrected − − i i 2 an error of the results of previous work on the analytic −√ x2 +i π ( im√ x2 +i) expression of the Feynman propagator in position − − − 2i 2i space of a Dirac spinor in (3 + 1)-dimensional space- = + = 0. (40) mπ(x2 i) mπ( x2 +i) time, and we have derived the generalized analytic − − 1582 Chinese Physics C (HEP & NP) Vol. 34 formulas of both the scalar Feynman propagator and This agreement supports our work and thus supports the spinor Feynman propagator in position space in our correction of Ref. [2] . The analytic formula of any (D + 1)-dimensional spacetime. The method we the spinor Feynman propagator in position space in have used in this paper is different from that used in arbitrary spacetime, obtained by us, is hitherto ab- Ref. [2]. And the result we have obtained shows that sent in literature so far as we know. According to the analytic formula of the Feynman propagator in Ref. [1], the Feynman propagator in position space position space can be also expressed in terms of Han- represents the energy due to the two external sources kel functions of the second kind and Modified Bessel located at two different points in spacetime. Figs. 1 functions in a general (D + 1)-dimensional space- and 2 show that in any dimensinal spacetime, the time, just like the known case in (3 + 1)-dimensional energy from two spacelike-separated external sources spacetime. From the obtained results, we have found decays exponentially with respect to their distance, an interesting recurrence relation among the spinor while the energy from two timelike-separated exter- Feynman propagator in (D + 1)-dimensional space- nal sources is dampedly oscillating with respect to time and the scalar Feynman propagators in (D+1)-, their distance. In both cases, the more the dimension (D 1)- and (D +3)-dimensional spacetime. The re- of spacetime, the more rapidly the energy decreases. − sult we have obtained for the scalar case agrees with the previous result of Ref. [11]. The equivalence of We would like to thank an anonymous referee for the two results was shown at the end of Section 2. very useful suggestions.

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